| L(s) = 1 | − 2·5-s − 4·7-s + 4·11-s − 2·13-s + 4·17-s − 4·19-s + 3·25-s − 12·31-s + 8·35-s + 12·41-s + 8·43-s − 4·47-s + 6·49-s + 12·53-s − 8·55-s + 12·59-s − 16·61-s + 4·65-s + 4·67-s + 4·71-s − 16·77-s − 12·83-s − 8·85-s − 12·89-s + 8·91-s + 8·95-s − 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 1.20·11-s − 0.554·13-s + 0.970·17-s − 0.917·19-s + 3/5·25-s − 2.15·31-s + 1.35·35-s + 1.87·41-s + 1.21·43-s − 0.583·47-s + 6/7·49-s + 1.64·53-s − 1.07·55-s + 1.56·59-s − 2.04·61-s + 0.496·65-s + 0.488·67-s + 0.474·71-s − 1.82·77-s − 1.31·83-s − 0.867·85-s − 1.27·89-s + 0.838·91-s + 0.820·95-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 136 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.44097652794206878449429596789, −7.12805855789313953568451573031, −6.84926743270669530148562633785, −6.70248116376451398078132182854, −6.21997110851803009590940904980, −5.74462990257915486030147031891, −5.53758073324190839372219145891, −5.37689620796515456532857994639, −4.40299743467775275273667889843, −4.37875710657195692891561441059, −3.96914210710921226615431669334, −3.74234041856073981185921520092, −3.16802261991586006136318858047, −3.07971686730757142127028287611, −2.35564144222474701133946999326, −2.16620481888116826376081203472, −1.22805125903995262707023634542, −1.03167932511855190552775937165, 0, 0,
1.03167932511855190552775937165, 1.22805125903995262707023634542, 2.16620481888116826376081203472, 2.35564144222474701133946999326, 3.07971686730757142127028287611, 3.16802261991586006136318858047, 3.74234041856073981185921520092, 3.96914210710921226615431669334, 4.37875710657195692891561441059, 4.40299743467775275273667889843, 5.37689620796515456532857994639, 5.53758073324190839372219145891, 5.74462990257915486030147031891, 6.21997110851803009590940904980, 6.70248116376451398078132182854, 6.84926743270669530148562633785, 7.12805855789313953568451573031, 7.44097652794206878449429596789
