| L(s) = 1 | − 1.80·2-s − 3-s + 1.24·4-s − 5-s + 1.80·6-s + 1.16·7-s + 1.36·8-s + 9-s + 1.80·10-s − 4.93·11-s − 1.24·12-s − 5.72·13-s − 2.10·14-s + 15-s − 4.94·16-s − 5.96·17-s − 1.80·18-s − 3.76·19-s − 1.24·20-s − 1.16·21-s + 8.88·22-s − 1.36·24-s + 25-s + 10.3·26-s − 27-s + 1.45·28-s + 10.2·29-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.622·4-s − 0.447·5-s + 0.735·6-s + 0.442·7-s + 0.480·8-s + 0.333·9-s + 0.569·10-s − 1.48·11-s − 0.359·12-s − 1.58·13-s − 0.562·14-s + 0.258·15-s − 1.23·16-s − 1.44·17-s − 0.424·18-s − 0.863·19-s − 0.278·20-s − 0.255·21-s + 1.89·22-s − 0.277·24-s + 0.200·25-s + 2.02·26-s − 0.192·27-s + 0.275·28-s + 1.90·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 7 | \( 1 - 1.16T + 7T^{2} \) |
| 11 | \( 1 + 4.93T + 11T^{2} \) |
| 13 | \( 1 + 5.72T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 9.40T + 47T^{2} \) |
| 53 | \( 1 - 0.718T + 53T^{2} \) |
| 59 | \( 1 - 6.18T + 59T^{2} \) |
| 61 | \( 1 - 4.15T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 9.42T + 71T^{2} \) |
| 73 | \( 1 - 9.34T + 73T^{2} \) |
| 79 | \( 1 + 7.30T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 3.77T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.72868591688492228243017265612, −6.92652118080677373115754975945, −6.49712196468935299554181267814, −5.18183883552218300186970334749, −4.77816802633398083644413066716, −4.20889361225025304294900761505, −2.55554171940893039232938693798, −2.23387033722361885823185204569, −0.77601691186777568395357533468, 0,
0.77601691186777568395357533468, 2.23387033722361885823185204569, 2.55554171940893039232938693798, 4.20889361225025304294900761505, 4.77816802633398083644413066716, 5.18183883552218300186970334749, 6.49712196468935299554181267814, 6.92652118080677373115754975945, 7.72868591688492228243017265612
