| L(s) = 1 | + 2-s + 4-s − 2·5-s + 4·7-s + 8-s − 2·10-s + 4·11-s + 13-s + 4·14-s + 16-s − 2·17-s − 8·19-s − 2·20-s + 4·22-s − 25-s + 26-s + 4·28-s − 6·29-s − 4·31-s + 32-s − 2·34-s − 8·35-s − 2·37-s − 8·38-s − 2·40-s + 10·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 1.83·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.196·26-s + 0.755·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.342·34-s − 1.35·35-s − 0.328·37-s − 1.29·38-s − 0.316·40-s + 1.56·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.837507996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837507996\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.06877268824115483606570314322, −11.29067241425549122912863548992, −10.82564350612112779643519160383, −9.025474627597816818050752077333, −8.129127522279784382432147914457, −7.16010307294570363491473287814, −5.93418826319229104786149140241, −4.49666099560716857332474018676, −3.92289433795456230197409104475, −1.87879273921165196191180493374,
1.87879273921165196191180493374, 3.92289433795456230197409104475, 4.49666099560716857332474018676, 5.93418826319229104786149140241, 7.16010307294570363491473287814, 8.129127522279784382432147914457, 9.025474627597816818050752077333, 10.82564350612112779643519160383, 11.29067241425549122912863548992, 12.06877268824115483606570314322
