| L(s) = 1 | − 2-s + 4-s − 5-s − 4.42·7-s − 8-s + 10-s + 2.28·11-s − 4.90·13-s + 4.42·14-s + 16-s + 4.42·17-s + 7.05·19-s − 20-s − 2.28·22-s + 0.622·23-s + 25-s + 4.90·26-s − 4.42·28-s + 2.76·29-s + 31-s − 32-s − 4.42·34-s + 4.42·35-s + 3.95·37-s − 7.05·38-s + 40-s − 3.67·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.67·7-s − 0.353·8-s + 0.316·10-s + 0.687·11-s − 1.35·13-s + 1.18·14-s + 0.250·16-s + 1.07·17-s + 1.61·19-s − 0.223·20-s − 0.486·22-s + 0.129·23-s + 0.200·25-s + 0.961·26-s − 0.836·28-s + 0.514·29-s + 0.179·31-s − 0.176·32-s − 0.759·34-s + 0.748·35-s + 0.650·37-s − 1.14·38-s + 0.158·40-s − 0.573·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 - 7.05T + 19T^{2} \) |
| 23 | \( 1 - 0.622T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 37 | \( 1 - 3.95T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 - 7.76T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 0.0459T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 + 8.29T + 67T^{2} \) |
| 71 | \( 1 + 2.75T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 9.95T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.514812387255335260242977278027, −7.41998800588847833120224210223, −7.23171573087337649389794716003, −6.28318451238150201519119834462, −5.53720272138750827817257994677, −4.38403318755440397466946723676, −3.20166229404660702336868870114, −2.89067512937629097428295378189, −1.21392566149952862064744667603, 0,
1.21392566149952862064744667603, 2.89067512937629097428295378189, 3.20166229404660702336868870114, 4.38403318755440397466946723676, 5.53720272138750827817257994677, 6.28318451238150201519119834462, 7.23171573087337649389794716003, 7.41998800588847833120224210223, 8.514812387255335260242977278027
