| L(s) = 1 | + 2-s + 4-s + 8-s − 11-s − 4·13-s + 16-s − 4·17-s − 22-s − 5·25-s − 4·26-s − 2·29-s − 4·31-s + 32-s − 4·34-s + 10·37-s + 12·41-s + 4·43-s − 44-s − 4·47-s − 5·50-s − 4·52-s + 10·53-s − 2·58-s + 4·59-s + 4·61-s − 4·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.301·11-s − 1.10·13-s + 1/4·16-s − 0.970·17-s − 0.213·22-s − 25-s − 0.784·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.685·34-s + 1.64·37-s + 1.87·41-s + 0.609·43-s − 0.150·44-s − 0.583·47-s − 0.707·50-s − 0.554·52-s + 1.37·53-s − 0.262·58-s + 0.520·59-s + 0.512·61-s − 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.689838023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.689838023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.56977491977112823145000115731, −7.00475118313187881954057066338, −6.18248323070809332233802961459, −5.61821639739805204865279478951, −4.86471741825285891630189726915, −4.24531381592818012353278806639, −3.57051680855412618231670981816, −2.40466364330865496923259480643, −2.20492528985927306506410301259, −0.66829490379841411848118262700,
0.66829490379841411848118262700, 2.20492528985927306506410301259, 2.40466364330865496923259480643, 3.57051680855412618231670981816, 4.24531381592818012353278806639, 4.86471741825285891630189726915, 5.61821639739805204865279478951, 6.18248323070809332233802961459, 7.00475118313187881954057066338, 7.56977491977112823145000115731
