| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 7-s − 3·8-s + 9-s + 4·11-s − 12-s + 2·13-s − 14-s − 16-s − 2·17-s + 18-s − 4·19-s − 21-s + 4·22-s − 3·24-s + 2·26-s + 27-s + 28-s + 29-s − 8·31-s + 5·32-s + 4·33-s − 2·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.218·21-s + 0.852·22-s − 0.612·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.185·29-s − 1.43·31-s + 0.883·32-s + 0.696·33-s − 0.342·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.25889865949143, −15.51010918237137, −15.01456682012133, −14.57769753643904, −14.14590876942099, −13.47197733481753, −13.04237999222505, −12.70792259679015, −11.87714726480633, −11.46413779804562, −10.66983759709733, −9.908257560399476, −9.399466681721698, −8.727363810197885, −8.597608558736440, −7.638205942779079, −6.764028062038135, −6.352224894194688, −5.714372968532732, −4.838493357856164, −4.169048428222337, −3.749489250533696, −3.118650138844831, −2.199681732242275, −1.250515423155074, 0,
1.250515423155074, 2.199681732242275, 3.118650138844831, 3.749489250533696, 4.169048428222337, 4.838493357856164, 5.714372968532732, 6.352224894194688, 6.764028062038135, 7.638205942779079, 8.597608558736440, 8.727363810197885, 9.399466681721698, 9.908257560399476, 10.66983759709733, 11.46413779804562, 11.87714726480633, 12.70792259679015, 13.04237999222505, 13.47197733481753, 14.14590876942099, 14.57769753643904, 15.01456682012133, 15.51010918237137, 16.25889865949143
