| L(s) = 1 | − 1.82·2-s + 1.31·4-s + 0.729·7-s + 1.24·8-s − 0.729·11-s − 3.38·13-s − 1.32·14-s − 4.90·16-s − 5.74·17-s + 6.11·19-s + 1.32·22-s − 9.48·23-s + 6.15·26-s + 0.957·28-s + 29-s + 5.48·31-s + 6.42·32-s + 10.4·34-s + 10.2·37-s − 11.1·38-s + 11.3·41-s + 10.1·43-s − 0.957·44-s + 17.2·46-s − 1.89·47-s − 6.46·49-s − 4.44·52-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.656·4-s + 0.275·7-s + 0.441·8-s − 0.219·11-s − 0.938·13-s − 0.354·14-s − 1.22·16-s − 1.39·17-s + 1.40·19-s + 0.282·22-s − 1.97·23-s + 1.20·26-s + 0.180·28-s + 0.185·29-s + 0.985·31-s + 1.13·32-s + 1.79·34-s + 1.69·37-s − 1.80·38-s + 1.76·41-s + 1.54·43-s − 0.144·44-s + 2.54·46-s − 0.276·47-s − 0.924·49-s − 0.616·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 7 | \( 1 - 0.729T + 7T^{2} \) |
| 11 | \( 1 + 0.729T + 11T^{2} \) |
| 13 | \( 1 + 3.38T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 - 6.11T + 19T^{2} \) |
| 23 | \( 1 + 9.48T + 23T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 1.89T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 - 2.55T + 67T^{2} \) |
| 71 | \( 1 - 4.83T + 71T^{2} \) |
| 73 | \( 1 + 6.29T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 - 0.0848T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 + 1.30T + 97T^{2} \) |
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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
−7.76078551745673623924494846169, −7.36360600747940387533298970826, −6.40421081687327639469882005116, −5.69897675700571338397776490793, −4.53575915322061060232162200423, −4.30554504411175952870095862997, −2.76651926476160236403664892133, −2.15494913471423604805004921847, −1.06415294287861087519602663419, 0,
1.06415294287861087519602663419, 2.15494913471423604805004921847, 2.76651926476160236403664892133, 4.30554504411175952870095862997, 4.53575915322061060232162200423, 5.69897675700571338397776490793, 6.40421081687327639469882005116, 7.36360600747940387533298970826, 7.76078551745673623924494846169
