L(s) = 1 | − 2.72·2-s + 3.06·3-s + 5.44·4-s − 8.35·6-s − 7-s − 9.40·8-s + 6.37·9-s + 1.29·11-s + 16.6·12-s + 6.43·13-s + 2.72·14-s + 14.7·16-s − 6.57·17-s − 17.3·18-s + 0.764·19-s − 3.06·21-s − 3.53·22-s − 23-s − 28.7·24-s − 17.5·26-s + 10.3·27-s − 5.44·28-s − 0.693·29-s + 4.00·31-s − 21.5·32-s + 3.96·33-s + 17.9·34-s + ⋯ |
L(s) = 1 | − 1.92·2-s + 1.76·3-s + 2.72·4-s − 3.41·6-s − 0.377·7-s − 3.32·8-s + 2.12·9-s + 0.390·11-s + 4.81·12-s + 1.78·13-s + 0.729·14-s + 3.69·16-s − 1.59·17-s − 4.09·18-s + 0.175·19-s − 0.667·21-s − 0.753·22-s − 0.208·23-s − 5.87·24-s − 3.44·26-s + 1.98·27-s − 1.02·28-s − 0.128·29-s + 0.718·31-s − 3.80·32-s + 0.690·33-s + 3.07·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.706394568\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.706394568\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 2.72T + 2T^{2} \) |
| 3 | \( 1 - 3.06T + 3T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 + 6.57T + 17T^{2} \) |
| 19 | \( 1 - 0.764T + 19T^{2} \) |
| 29 | \( 1 + 0.693T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 - 3.82T + 43T^{2} \) |
| 47 | \( 1 - 0.878T + 47T^{2} \) |
| 53 | \( 1 + 6.13T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 + 9.80T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 - 6.97T + 71T^{2} \) |
| 73 | \( 1 - 2.25T + 73T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 1.27T + 97T^{2} \) |
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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
−8.618564684369883384424274739807, −8.008075123899414547259855364555, −7.42574280587544769600812777007, −6.53607597155846052583879341791, −6.16133359459681963989813425553, −4.21671002144235429080913744571, −3.40801734109051404591489063005, −2.60448617202244257482880673648, −1.87511823594259683760597261015, −0.950718520206872078427041566477,
0.950718520206872078427041566477, 1.87511823594259683760597261015, 2.60448617202244257482880673648, 3.40801734109051404591489063005, 4.21671002144235429080913744571, 6.16133359459681963989813425553, 6.53607597155846052583879341791, 7.42574280587544769600812777007, 8.008075123899414547259855364555, 8.618564684369883384424274739807