| L(s) = 1 | − 2-s + 4-s + 1.07·5-s + 3.48·7-s − 8-s − 1.07·10-s − 11-s − 3.25·13-s − 3.48·14-s + 16-s − 1.76·17-s + 5.05·19-s + 1.07·20-s + 22-s + 2.12·23-s − 3.84·25-s + 3.25·26-s + 3.48·28-s − 7.17·29-s + 2·31-s − 32-s + 1.76·34-s + 3.73·35-s + 10.8·37-s − 5.05·38-s − 1.07·40-s − 0.256·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.480·5-s + 1.31·7-s − 0.353·8-s − 0.339·10-s − 0.301·11-s − 0.902·13-s − 0.930·14-s + 0.250·16-s − 0.429·17-s + 1.15·19-s + 0.240·20-s + 0.213·22-s + 0.442·23-s − 0.769·25-s + 0.638·26-s + 0.658·28-s − 1.33·29-s + 0.359·31-s − 0.176·32-s + 0.303·34-s + 0.631·35-s + 1.78·37-s − 0.819·38-s − 0.169·40-s − 0.0400·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.777580415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.777580415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 5 | \( 1 - 1.07T + 5T^{2} \) |
| 7 | \( 1 - 3.48T + 7T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 - 2.12T + 23T^{2} \) |
| 29 | \( 1 + 7.17T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 0.256T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 + 6.35T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 - 5.69T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 2.57T + 79T^{2} \) |
| 83 | \( 1 + 1.84T + 83T^{2} \) |
| 89 | \( 1 - 1.36T + 89T^{2} \) |
| 97 | \( 1 - 2.54T + 97T^{2} \) |
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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
−7.85298407935592606201291254346, −7.23621583009655387796934804810, −6.48687742228024750685178763797, −5.52219168115123180044142517355, −5.13937388646515068767343767924, −4.33556494244768613777424778967, −3.25771450011585378947398454557, −2.27863183755254424521265399060, −1.78131726932644806865272018024, −0.71719329301127853796775876922,
0.71719329301127853796775876922, 1.78131726932644806865272018024, 2.27863183755254424521265399060, 3.25771450011585378947398454557, 4.33556494244768613777424778967, 5.13937388646515068767343767924, 5.52219168115123180044142517355, 6.48687742228024750685178763797, 7.23621583009655387796934804810, 7.85298407935592606201291254346
