| L(s) = 1 | + 7·3-s − 5·5-s + 2·7-s + 22·9-s + 37·11-s − 27·13-s − 35·15-s + 24·17-s − 88·19-s + 14·21-s + 28·23-s − 100·25-s − 35·27-s + 29·29-s + 143·31-s + 259·33-s − 10·35-s + 360·37-s − 189·39-s + 386·41-s + 381·43-s − 110·45-s + 103·47-s − 339·49-s + 168·51-s + 431·53-s − 185·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 0.447·5-s + 0.107·7-s + 0.814·9-s + 1.01·11-s − 0.576·13-s − 0.602·15-s + 0.342·17-s − 1.06·19-s + 0.145·21-s + 0.253·23-s − 4/5·25-s − 0.249·27-s + 0.185·29-s + 0.828·31-s + 1.36·33-s − 0.0482·35-s + 1.59·37-s − 0.776·39-s + 1.47·41-s + 1.35·43-s − 0.364·45-s + 0.319·47-s − 0.988·49-s + 0.461·51-s + 1.11·53-s − 0.453·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.539402962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.539402962\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 5 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 37 T + p^{3} T^{2} \) |
| 13 | \( 1 + 27 T + p^{3} T^{2} \) |
| 17 | \( 1 - 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 88 T + p^{3} T^{2} \) |
| 23 | \( 1 - 28 T + p^{3} T^{2} \) |
| 31 | \( 1 - 143 T + p^{3} T^{2} \) |
| 37 | \( 1 - 360 T + p^{3} T^{2} \) |
| 41 | \( 1 - 386 T + p^{3} T^{2} \) |
| 43 | \( 1 - 381 T + p^{3} T^{2} \) |
| 47 | \( 1 - 103 T + p^{3} T^{2} \) |
| 53 | \( 1 - 431 T + p^{3} T^{2} \) |
| 59 | \( 1 - 288 T + p^{3} T^{2} \) |
| 61 | \( 1 - 840 T + p^{3} T^{2} \) |
| 67 | \( 1 + 180 T + p^{3} T^{2} \) |
| 71 | \( 1 + 706 T + p^{3} T^{2} \) |
| 73 | \( 1 - 716 T + p^{3} T^{2} \) |
| 79 | \( 1 + 931 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 642 T + p^{3} T^{2} \) |
| 97 | \( 1 - 486 T + p^{3} T^{2} \) |
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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
−8.870707852987378592638247938105, −8.086638611644483166040358116262, −7.59330996550778218133344256801, −6.67475446269757354770951093105, −5.74154068336114673510948000112, −4.32353605657413057126080181358, −3.97503325832271973647303837011, −2.84254233945346869397993539508, −2.12972351725539975655583383911, −0.827894620932734192572087134526,
0.827894620932734192572087134526, 2.12972351725539975655583383911, 2.84254233945346869397993539508, 3.97503325832271973647303837011, 4.32353605657413057126080181358, 5.74154068336114673510948000112, 6.67475446269757354770951093105, 7.59330996550778218133344256801, 8.086638611644483166040358116262, 8.870707852987378592638247938105
