L(s) = 1 | − 2.70·3-s + 2.87·5-s + 4.19·7-s + 4.32·9-s − 3.32·11-s − 4.90·13-s − 7.77·15-s + 2·17-s + 7.15·19-s − 11.3·21-s + 5.15·23-s + 3.25·25-s − 3.57·27-s + 3.49·29-s − 9.26·31-s + 8.99·33-s + 12.0·35-s + 37-s + 13.2·39-s + 7.35·41-s + 1.74·43-s + 12.4·45-s + 10.9·47-s + 10.6·49-s − 5.41·51-s − 0.784·53-s − 9.54·55-s + ⋯ |
L(s) = 1 | − 1.56·3-s + 1.28·5-s + 1.58·7-s + 1.44·9-s − 1.00·11-s − 1.35·13-s − 2.00·15-s + 0.485·17-s + 1.64·19-s − 2.47·21-s + 1.07·23-s + 0.651·25-s − 0.688·27-s + 0.648·29-s − 1.66·31-s + 1.56·33-s + 2.03·35-s + 0.164·37-s + 2.12·39-s + 1.14·41-s + 0.266·43-s + 1.85·45-s + 1.59·47-s + 1.51·49-s − 0.757·51-s − 0.107·53-s − 1.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.242292292\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.242292292\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 + 2.70T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 11 | \( 1 + 3.32T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 7.15T + 19T^{2} \) |
| 23 | \( 1 - 5.15T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 41 | \( 1 - 7.35T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 0.784T + 53T^{2} \) |
| 59 | \( 1 + 0.513T + 59T^{2} \) |
| 61 | \( 1 - 4.62T + 61T^{2} \) |
| 67 | \( 1 - 6.28T + 67T^{2} \) |
| 71 | \( 1 + 5.27T + 71T^{2} \) |
| 73 | \( 1 + 2.17T + 73T^{2} \) |
| 79 | \( 1 - 7.97T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 4.76T + 89T^{2} \) |
| 97 | \( 1 + 0.431T + 97T^{2} \) |
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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
−10.75673395738502661533582263022, −10.06842177232324192424250110128, −9.210115680807869111504505487189, −7.70909005934606383691505664693, −7.12041094245226010103759750742, −5.57319542401709486210886403294, −5.42610456724186368189892415129, −4.70816959737180756415677472397, −2.46586028006452514460070658559, −1.14988451018935665019834727923,
1.14988451018935665019834727923, 2.46586028006452514460070658559, 4.70816959737180756415677472397, 5.42610456724186368189892415129, 5.57319542401709486210886403294, 7.12041094245226010103759750742, 7.70909005934606383691505664693, 9.210115680807869111504505487189, 10.06842177232324192424250110128, 10.75673395738502661533582263022