| L(s) = 1 | − 0.788·2-s − 1.37·4-s − 4.52·7-s + 2.66·8-s + 3.48·11-s + 5.61·13-s + 3.57·14-s + 0.652·16-s − 3.78·17-s − 1.21·19-s − 2.74·22-s − 4.09·23-s − 4.43·26-s + 6.23·28-s − 4.02·29-s − 2.48·31-s − 5.84·32-s + 2.98·34-s − 37-s + 0.955·38-s + 12.6·41-s − 0.120·43-s − 4.79·44-s + 3.22·46-s − 4.57·47-s + 13.5·49-s − 7.73·52-s + ⋯ |
| L(s) = 1 | − 0.557·2-s − 0.688·4-s − 1.71·7-s + 0.942·8-s + 1.05·11-s + 1.55·13-s + 0.954·14-s + 0.163·16-s − 0.918·17-s − 0.277·19-s − 0.585·22-s − 0.852·23-s − 0.869·26-s + 1.17·28-s − 0.746·29-s − 0.445·31-s − 1.03·32-s + 0.512·34-s − 0.164·37-s + 0.154·38-s + 1.97·41-s − 0.0184·43-s − 0.723·44-s + 0.475·46-s − 0.667·47-s + 1.92·49-s − 1.07·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.788T + 2T^{2} \) |
| 7 | \( 1 + 4.52T + 7T^{2} \) |
| 11 | \( 1 - 3.48T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 + 3.78T + 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 41 | \( 1 - 12.6T + 41T^{2} \) |
| 43 | \( 1 + 0.120T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 + 4.83T + 53T^{2} \) |
| 59 | \( 1 - 0.709T + 59T^{2} \) |
| 61 | \( 1 - 8.77T + 61T^{2} \) |
| 67 | \( 1 + 8.01T + 67T^{2} \) |
| 71 | \( 1 - 2.98T + 71T^{2} \) |
| 73 | \( 1 - 2.05T + 73T^{2} \) |
| 79 | \( 1 + 1.46T + 79T^{2} \) |
| 83 | \( 1 - 6.74T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−7.51351407401193260004797246465, −6.64811688178617433599513930268, −6.24092737114813718477463164281, −5.60418510028081107259217676532, −4.33770217287271494028134243667, −3.87551628035093539150178902773, −3.32232043276818001516107846792, −2.04796953968114136694855547342, −0.992068726530475426815756744089, 0,
0.992068726530475426815756744089, 2.04796953968114136694855547342, 3.32232043276818001516107846792, 3.87551628035093539150178902773, 4.33770217287271494028134243667, 5.60418510028081107259217676532, 6.24092737114813718477463164281, 6.64811688178617433599513930268, 7.51351407401193260004797246465
