| L(s) = 1 | + i·3-s − i·7-s + 2·9-s + 3·11-s − i·13-s + 3i·17-s − 2·19-s + 21-s − 6i·23-s + 5i·27-s + 9·29-s + 8·31-s + 3i·33-s + 10i·37-s + 39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.377i·7-s + 0.666·9-s + 0.904·11-s − 0.277i·13-s + 0.727i·17-s − 0.458·19-s + 0.218·21-s − 1.25i·23-s + 0.962i·27-s + 1.67·29-s + 1.43·31-s + 0.522i·33-s + 1.64i·37-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64136 + 0.387474i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64136 + 0.387474i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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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
−10.24262838775023461861404100231, −9.976166749486714071219095425533, −8.760860071692623192050578806776, −8.096563930354053160040327527510, −6.79620981619347794681097174174, −6.25893035378369345296372846428, −4.68562734852391706233306225960, −4.23813789251218548546675402166, −2.98065680793804463112484952603, −1.27544196335232918275451230676,
1.18393045158134296854279413414, 2.45294574152214557898252706760, 3.87379750467949118401159901989, 4.88315657169905388326817582719, 6.14552220120859690071768519404, 6.84705877854602587425467221678, 7.67130170734319708606074022464, 8.688367510441189518248350587787, 9.490358204226610368657489955792, 10.29150897846596281766823303888
