| L(s) = 1 | − 4-s + 4·5-s − 9-s + 4·11-s + 16-s − 4·19-s − 4·20-s + 11·25-s + 4·29-s − 8·31-s + 36-s − 12·41-s − 4·44-s − 4·45-s − 2·49-s + 16·55-s + 28·59-s + 20·61-s − 64-s + 16·71-s + 4·76-s + 16·79-s + 4·80-s + 81-s + 36·89-s − 16·95-s − 4·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.78·5-s − 1/3·9-s + 1.20·11-s + 1/4·16-s − 0.917·19-s − 0.894·20-s + 11/5·25-s + 0.742·29-s − 1.43·31-s + 1/6·36-s − 1.87·41-s − 0.603·44-s − 0.596·45-s − 2/7·49-s + 2.15·55-s + 3.64·59-s + 2.56·61-s − 1/8·64-s + 1.89·71-s + 0.458·76-s + 1.80·79-s + 0.447·80-s + 1/9·81-s + 3.81·89-s − 1.64·95-s − 0.402·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.106859932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106859932\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.70541008195990028035299899017, −10.95030909764592178474522169338, −10.44950589932283994661408709602, −10.23845136228507693970389687210, −9.475364833362792739908843742432, −9.453993206689234124581425350749, −8.927157072855751300511983482411, −8.323560979338947872027091728120, −8.199621978548240832095296968422, −7.02555179057765223695230575436, −6.59800822985621947604709322250, −6.52567964516979694630917506526, −5.69097944702842112330053025984, −5.19510855767203085831627464049, −4.99600095108582412245908631971, −3.78286145293792027194094329732, −3.72794716259632351621374002454, −2.44101128541060251824225185472, −2.04586530366640465688845444023, −1.06042640914004630859042630322,
1.06042640914004630859042630322, 2.04586530366640465688845444023, 2.44101128541060251824225185472, 3.72794716259632351621374002454, 3.78286145293792027194094329732, 4.99600095108582412245908631971, 5.19510855767203085831627464049, 5.69097944702842112330053025984, 6.52567964516979694630917506526, 6.59800822985621947604709322250, 7.02555179057765223695230575436, 8.199621978548240832095296968422, 8.323560979338947872027091728120, 8.927157072855751300511983482411, 9.453993206689234124581425350749, 9.475364833362792739908843742432, 10.23845136228507693970389687210, 10.44950589932283994661408709602, 10.95030909764592178474522169338, 11.70541008195990028035299899017
