| L(s) = 1 | + 64·4-s − 306·9-s + 3.07e3·16-s − 2.10e4·19-s + 9.15e4·31-s − 1.95e4·36-s + 4.70e5·49-s − 2.50e5·61-s + 1.31e5·64-s − 1.34e6·76-s − 1.36e6·79-s − 4.37e5·81-s − 2.60e6·109-s + 7.08e6·121-s + 5.86e6·124-s + 127-s + 131-s + 137-s + 139-s − 9.40e5·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.90e6·169-s + 6.43e6·171-s + ⋯ |
| L(s) = 1 | + 4-s − 0.419·9-s + 3/4·16-s − 3.06·19-s + 3.07·31-s − 0.419·36-s + 3.99·49-s − 1.10·61-s + 1/2·64-s − 3.06·76-s − 2.76·79-s − 0.823·81-s − 2.01·109-s + 3.99·121-s + 3.07·124-s − 0.314·144-s + 0.394·169-s + 1.28·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8716325303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8716325303\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 34 p^{2} T^{2} + p^{12} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 235294 T^{2} + p^{12} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 3541970 T^{2} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 951118 T^{2} + p^{12} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 28202690 T^{2} + p^{12} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5258 T + p^{6} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 191004770 T^{2} + p^{12} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1184779442 T^{2} + p^{12} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 22898 T + p^{6} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 3971505454 T^{2} + p^{12} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 9219079010 T^{2} + p^{12} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 12601689262 T^{2} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 10801249342 T^{2} + p^{12} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 7253988050 T^{2} + p^{12} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 22449655150 T^{2} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 62566 T + p^{6} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 11539170866 T^{2} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 251546372642 T^{2} + p^{12} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 230976407522 T^{2} + p^{12} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 340562 T + p^{6} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 407613512306 T^{2} + p^{12} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 844406214050 T^{2} + p^{12} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1586934670462 T^{2} + p^{12} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.476820434472981727535919426101, −8.203879704055503590636255930441, −7.77431207257399026891413414500, −7.43128589632575662611403432942, −7.31292530974798972108424608486, −6.98875253412587546327836403798, −6.50878272372524157608700327732, −6.26022745896911096392962710220, −6.22888631907273585976186891932, −6.03572791941070882267950991620, −5.42191476091708382183590940833, −5.27275761593684370852909798591, −4.77550169945344862935320375631, −4.22870332500872164175966606879, −4.18696767546533948902362758781, −4.06935168054215949448167112089, −3.34604728106528490016635703666, −2.89364106199141203766927317079, −2.53506976120086073910876760097, −2.48386441672522719088108839671, −2.09561511829983640972034392758, −1.50088773631870837053541255559, −1.15234416797797178852660397689, −0.73103366940431477691553198266, −0.12488629094379360870784103059,
0.12488629094379360870784103059, 0.73103366940431477691553198266, 1.15234416797797178852660397689, 1.50088773631870837053541255559, 2.09561511829983640972034392758, 2.48386441672522719088108839671, 2.53506976120086073910876760097, 2.89364106199141203766927317079, 3.34604728106528490016635703666, 4.06935168054215949448167112089, 4.18696767546533948902362758781, 4.22870332500872164175966606879, 4.77550169945344862935320375631, 5.27275761593684370852909798591, 5.42191476091708382183590940833, 6.03572791941070882267950991620, 6.22888631907273585976186891932, 6.26022745896911096392962710220, 6.50878272372524157608700327732, 6.98875253412587546327836403798, 7.31292530974798972108424608486, 7.43128589632575662611403432942, 7.77431207257399026891413414500, 8.203879704055503590636255930441, 8.476820434472981727535919426101
