| L(s) = 1 | − 2·2-s − 7·3-s + 4·4-s + 14·6-s − 7·7-s − 8·8-s + 22·9-s − 33·11-s − 28·12-s + 43·13-s + 14·14-s + 16·16-s − 111·17-s − 44·18-s − 70·19-s + 49·21-s + 66·22-s − 42·23-s + 56·24-s − 86·26-s + 35·27-s − 28·28-s − 225·29-s − 88·31-s − 32·32-s + 231·33-s + 222·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·3-s + 1/2·4-s + 0.952·6-s − 0.377·7-s − 0.353·8-s + 0.814·9-s − 0.904·11-s − 0.673·12-s + 0.917·13-s + 0.267·14-s + 1/4·16-s − 1.58·17-s − 0.576·18-s − 0.845·19-s + 0.509·21-s + 0.639·22-s − 0.380·23-s + 0.476·24-s − 0.648·26-s + 0.249·27-s − 0.188·28-s − 1.44·29-s − 0.509·31-s − 0.176·32-s + 1.21·33-s + 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4092975955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4092975955\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 43 T + p^{3} T^{2} \) |
| 17 | \( 1 + 111 T + p^{3} T^{2} \) |
| 19 | \( 1 + 70 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 225 T + p^{3} T^{2} \) |
| 31 | \( 1 + 88 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 432 T + p^{3} T^{2} \) |
| 43 | \( 1 - 178 T + p^{3} T^{2} \) |
| 47 | \( 1 + 411 T + p^{3} T^{2} \) |
| 53 | \( 1 - 708 T + p^{3} T^{2} \) |
| 59 | \( 1 - 480 T + p^{3} T^{2} \) |
| 61 | \( 1 - 812 T + p^{3} T^{2} \) |
| 67 | \( 1 + 596 T + p^{3} T^{2} \) |
| 71 | \( 1 - 432 T + p^{3} T^{2} \) |
| 73 | \( 1 - 358 T + p^{3} T^{2} \) |
| 79 | \( 1 - 425 T + p^{3} T^{2} \) |
| 83 | \( 1 + 972 T + p^{3} T^{2} \) |
| 89 | \( 1 - 960 T + p^{3} T^{2} \) |
| 97 | \( 1 - 709 T + p^{3} T^{2} \) |
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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
−10.98311330400093551301851212801, −10.44046612362465010040766187151, −9.264081560414778381516269845729, −8.321809439669499203278494486639, −7.08668287137495386067353253497, −6.23754661485049705750952470360, −5.46604885394831259230103580999, −4.07294704593226627937467285318, −2.24134013645726818640515232524, −0.48688644099293187024454329477,
0.48688644099293187024454329477, 2.24134013645726818640515232524, 4.07294704593226627937467285318, 5.46604885394831259230103580999, 6.23754661485049705750952470360, 7.08668287137495386067353253497, 8.321809439669499203278494486639, 9.264081560414778381516269845729, 10.44046612362465010040766187151, 10.98311330400093551301851212801
