| L(s) = 1 | − 5-s − 7-s + 11-s − 13-s − 5·19-s + 2·23-s − 4·25-s + 29-s − 8·31-s + 35-s + 37-s − 6·43-s + 47-s + 49-s + 2·53-s − 55-s + 9·59-s + 10·61-s + 65-s − 7·67-s + 9·73-s − 77-s + 6·89-s + 91-s + 5·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.277·13-s − 1.14·19-s + 0.417·23-s − 4/5·25-s + 0.185·29-s − 1.43·31-s + 0.169·35-s + 0.164·37-s − 0.914·43-s + 0.145·47-s + 1/7·49-s + 0.274·53-s − 0.134·55-s + 1.17·59-s + 1.28·61-s + 0.124·65-s − 0.855·67-s + 1.05·73-s − 0.113·77-s + 0.635·89-s + 0.104·91-s + 0.512·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.172828009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.172828009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.51862880957925, −15.94571897839169, −15.31558036646825, −14.76973824579180, −14.42315314889173, −13.43732666416010, −13.15243909798626, −12.38958755149540, −11.98765971770620, −11.21078573705696, −10.81655087659533, −9.984906711126447, −9.541503286956012, −8.724982968912640, −8.322200145536922, −7.462543756373324, −6.959377320589358, −6.290675967835189, −5.572002605651282, −4.820252102963109, −3.985945882762662, −3.546839971940087, −2.530452313870118, −1.756964484304600, −0.4882945924852430,
0.4882945924852430, 1.756964484304600, 2.530452313870118, 3.546839971940087, 3.985945882762662, 4.820252102963109, 5.572002605651282, 6.290675967835189, 6.959377320589358, 7.462543756373324, 8.322200145536922, 8.724982968912640, 9.541503286956012, 9.984906711126447, 10.81655087659533, 11.21078573705696, 11.98765971770620, 12.38958755149540, 13.15243909798626, 13.43732666416010, 14.42315314889173, 14.76973824579180, 15.31558036646825, 15.94571897839169, 16.51862880957925
