| L(s) = 1 | + 2.11·2-s + 1.84·3-s + 2.49·4-s − 2.40·5-s + 3.90·6-s + 1.04·8-s + 0.388·9-s − 5.10·10-s + 5.87·11-s + 4.58·12-s + 6.24·13-s − 4.43·15-s − 2.77·16-s + 5.42·17-s + 0.823·18-s + 2.23·19-s − 6.00·20-s + 12.4·22-s − 23-s + 1.92·24-s + 0.804·25-s + 13.2·26-s − 4.80·27-s + 0.642·29-s − 9.39·30-s − 7.84·31-s − 7.96·32-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.06·3-s + 1.24·4-s − 1.07·5-s + 1.59·6-s + 0.368·8-s + 0.129·9-s − 1.61·10-s + 1.77·11-s + 1.32·12-s + 1.73·13-s − 1.14·15-s − 0.693·16-s + 1.31·17-s + 0.193·18-s + 0.513·19-s − 1.34·20-s + 2.65·22-s − 0.208·23-s + 0.391·24-s + 0.160·25-s + 2.59·26-s − 0.925·27-s + 0.119·29-s − 1.71·30-s − 1.40·31-s − 1.40·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.662126322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.662126322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 11 | \( 1 - 5.87T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 - 5.42T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 29 | \( 1 - 0.642T + 29T^{2} \) |
| 31 | \( 1 + 7.84T + 31T^{2} \) |
| 37 | \( 1 - 0.557T + 37T^{2} \) |
| 41 | \( 1 + 2.56T + 41T^{2} \) |
| 43 | \( 1 + 8.81T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 - 3.01T + 53T^{2} \) |
| 59 | \( 1 + 4.17T + 59T^{2} \) |
| 61 | \( 1 - 0.148T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 7.93T + 71T^{2} \) |
| 73 | \( 1 + 4.28T + 73T^{2} \) |
| 79 | \( 1 + 0.861T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 + 6.32T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−9.647190212771186392214878834594, −8.818700740725425936136041433303, −8.204061891780063420460579647258, −7.23057555726776718015045592043, −6.31914287913814390853667046698, −5.48438725358063105554042675493, −4.14098606541439908119826587758, −3.57441763060301435432091247418, −3.29331623962843159099829708557, −1.56157244006441539728510786452,
1.56157244006441539728510786452, 3.29331623962843159099829708557, 3.57441763060301435432091247418, 4.14098606541439908119826587758, 5.48438725358063105554042675493, 6.31914287913814390853667046698, 7.23057555726776718015045592043, 8.204061891780063420460579647258, 8.818700740725425936136041433303, 9.647190212771186392214878834594
