L(s) = 1 | − 3-s + 4-s − 11-s − 12-s − 13-s + 16-s − 19-s − 23-s + 25-s + 27-s + 2·29-s + 33-s − 37-s + 39-s − 41-s − 44-s + 2·47-s − 48-s + 49-s − 52-s − 53-s + 57-s − 61-s + 64-s + 69-s − 75-s − 76-s + ⋯ |
L(s) = 1 | − 3-s + 4-s − 11-s − 12-s − 13-s + 16-s − 19-s − 23-s + 25-s + 27-s + 2·29-s + 33-s − 37-s + 39-s − 41-s − 44-s + 2·47-s − 48-s + 49-s − 52-s − 53-s + 57-s − 61-s + 64-s + 69-s − 75-s − 76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5030252953\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5030252953\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 107 | \( 1 - T \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−14.04011337995081042829635922421, −12.41714126767367420427897494631, −12.04504638430051238311412403063, −10.71867685031617401459794955967, −10.31805535325114251598975346150, −8.373724445978545223480806975034, −7.08658080864405272708817928649, −6.07457868538096646079651191674, −4.91802014171830707927308424854, −2.61608028997495214067096557530,
2.61608028997495214067096557530, 4.91802014171830707927308424854, 6.07457868538096646079651191674, 7.08658080864405272708817928649, 8.373724445978545223480806975034, 10.31805535325114251598975346150, 10.71867685031617401459794955967, 12.04504638430051238311412403063, 12.41714126767367420427897494631, 14.04011337995081042829635922421