| L(s) = 1 | − 9·3-s − 25·5-s − 12·7-s + 81·9-s − 112·11-s − 974·13-s + 225·15-s + 2.18e3·17-s − 1.42e3·19-s + 108·21-s − 3.21e3·23-s + 625·25-s − 729·27-s − 4.15e3·29-s + 5.68e3·31-s + 1.00e3·33-s + 300·35-s + 6.48e3·37-s + 8.76e3·39-s + 5.40e3·41-s + 2.17e4·43-s − 2.02e3·45-s + 368·47-s − 1.66e4·49-s − 1.96e4·51-s + 1.25e4·53-s + 2.80e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.0925·7-s + 1/3·9-s − 0.279·11-s − 1.59·13-s + 0.258·15-s + 1.83·17-s − 0.902·19-s + 0.0534·21-s − 1.26·23-s + 1/5·25-s − 0.192·27-s − 0.916·29-s + 1.06·31-s + 0.161·33-s + 0.0413·35-s + 0.778·37-s + 0.922·39-s + 0.501·41-s + 1.79·43-s − 0.149·45-s + 0.0242·47-s − 0.991·49-s − 1.05·51-s + 0.615·53-s + 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.070240117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.070240117\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 12 T + p^{5} T^{2} \) |
| 11 | \( 1 + 112 T + p^{5} T^{2} \) |
| 13 | \( 1 + 974 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2182 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1420 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3216 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4150 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5688 T + p^{5} T^{2} \) |
| 37 | \( 1 - 6482 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5402 T + p^{5} T^{2} \) |
| 43 | \( 1 - 21764 T + p^{5} T^{2} \) |
| 47 | \( 1 - 368 T + p^{5} T^{2} \) |
| 53 | \( 1 - 12586 T + p^{5} T^{2} \) |
| 59 | \( 1 - 25520 T + p^{5} T^{2} \) |
| 61 | \( 1 - 11782 T + p^{5} T^{2} \) |
| 67 | \( 1 - 13188 T + p^{5} T^{2} \) |
| 71 | \( 1 - 35968 T + p^{5} T^{2} \) |
| 73 | \( 1 - 73186 T + p^{5} T^{2} \) |
| 79 | \( 1 - 52440 T + p^{5} T^{2} \) |
| 83 | \( 1 + 69036 T + p^{5} T^{2} \) |
| 89 | \( 1 + 33870 T + p^{5} T^{2} \) |
| 97 | \( 1 - 143042 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.37480922618868530028618503009, −10.22903406265139982070509205916, −9.628405240423155599792742736573, −8.046239626030600083526960429174, −7.39916524139214647034106145530, −6.09759029310649678388244851924, −5.07713546048688053435543968067, −3.92877909561026773161213598813, −2.39422878319756951955346415178, −0.61306123469931468529535963010,
0.61306123469931468529535963010, 2.39422878319756951955346415178, 3.92877909561026773161213598813, 5.07713546048688053435543968067, 6.09759029310649678388244851924, 7.39916524139214647034106145530, 8.046239626030600083526960429174, 9.628405240423155599792742736573, 10.22903406265139982070509205916, 11.37480922618868530028618503009
