| L(s) = 1 | + 3·2-s + 4·4-s + 3·8-s + 6·11-s − 2·13-s + 3·16-s + 6·17-s + 6·19-s + 18·22-s + 12·23-s − 5·25-s − 6·26-s + 10·31-s + 6·32-s + 18·34-s − 4·37-s + 18·38-s − 16·43-s + 24·44-s + 36·46-s + 6·47-s − 15·50-s − 8·52-s + 6·53-s − 6·59-s + 6·61-s + 30·62-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s + 1.06·8-s + 1.80·11-s − 0.554·13-s + 3/4·16-s + 1.45·17-s + 1.37·19-s + 3.83·22-s + 2.50·23-s − 25-s − 1.17·26-s + 1.79·31-s + 1.06·32-s + 3.08·34-s − 0.657·37-s + 2.91·38-s − 2.43·43-s + 3.61·44-s + 5.30·46-s + 0.875·47-s − 2.12·50-s − 1.10·52-s + 0.824·53-s − 0.781·59-s + 0.768·61-s + 3.81·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32867289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32867289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.84815447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.84815447\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 95 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 173 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.270612109086063232287708959359, −7.69629757132774707611651376452, −7.39496068555586426443115287178, −7.17955611398549923853908406109, −6.62305805131218474681205173489, −6.52839518284144799122193282594, −5.86322008664896671218046157369, −5.81398570354841287006827766299, −5.17501842036711965255703147720, −5.04922240682503243801843755772, −4.69777243122186607431825797567, −4.40802129489538029928248668667, −3.79572697239028377233237807751, −3.50998178364644846807811630612, −3.16810263603139382652102846667, −3.11108144127331549652676534880, −2.31518502414556948623382678579, −1.70417661669501108230058458565, −1.05093587368298752338815604664, −0.883430952743246814743964384621,
0.883430952743246814743964384621, 1.05093587368298752338815604664, 1.70417661669501108230058458565, 2.31518502414556948623382678579, 3.11108144127331549652676534880, 3.16810263603139382652102846667, 3.50998178364644846807811630612, 3.79572697239028377233237807751, 4.40802129489538029928248668667, 4.69777243122186607431825797567, 5.04922240682503243801843755772, 5.17501842036711965255703147720, 5.81398570354841287006827766299, 5.86322008664896671218046157369, 6.52839518284144799122193282594, 6.62305805131218474681205173489, 7.17955611398549923853908406109, 7.39496068555586426443115287178, 7.69629757132774707611651376452, 8.270612109086063232287708959359
