| L(s) = 1 | − 0.786·3-s − 3.29·5-s − 2.38·9-s + 1.29·11-s − 1.21·13-s + 2.59·15-s − 4.08·17-s + 19-s − 8.95·23-s + 5.87·25-s + 4.23·27-s − 9.38·29-s − 1.02·33-s − 2·37-s + 0.954·39-s − 3.57·41-s + 7.72·43-s + 7.85·45-s − 9.46·47-s + 3.21·51-s − 11.9·53-s − 4.27·55-s − 0.786·57-s + 7.21·59-s − 4.87·61-s + 3.99·65-s + 11.3·67-s + ⋯ |
| L(s) = 1 | − 0.454·3-s − 1.47·5-s − 0.793·9-s + 0.391·11-s − 0.336·13-s + 0.669·15-s − 0.990·17-s + 0.229·19-s − 1.86·23-s + 1.17·25-s + 0.814·27-s − 1.74·29-s − 0.177·33-s − 0.328·37-s + 0.152·39-s − 0.558·41-s + 1.17·43-s + 1.17·45-s − 1.38·47-s + 0.449·51-s − 1.64·53-s − 0.576·55-s − 0.104·57-s + 0.939·59-s − 0.623·61-s + 0.496·65-s + 1.39·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1706507673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1706507673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.786T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 + 1.21T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 23 | \( 1 + 8.95T + 23T^{2} \) |
| 29 | \( 1 + 9.38T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 3.57T + 41T^{2} \) |
| 43 | \( 1 - 7.72T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 - 9.57T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 8.59T + 97T^{2} \) |
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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
−7.974531417152484732717604997710, −7.23568322917651081978876098659, −6.54036194293671975182006838754, −5.81778228855105169601843295063, −5.04661974617232001466806677259, −4.19091412997547462066225798548, −3.74876378024051065381116480851, −2.81941899635224422438527585033, −1.74353655445254116995674239693, −0.20408656993888991514899041791,
0.20408656993888991514899041791, 1.74353655445254116995674239693, 2.81941899635224422438527585033, 3.74876378024051065381116480851, 4.19091412997547462066225798548, 5.04661974617232001466806677259, 5.81778228855105169601843295063, 6.54036194293671975182006838754, 7.23568322917651081978876098659, 7.974531417152484732717604997710
