| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 7-s − 2·10-s + 5·11-s + 3·13-s + 2·14-s − 4·16-s + 5·17-s + 2·20-s − 10·22-s + 23-s + 25-s − 6·26-s − 2·28-s − 3·29-s + 6·31-s + 8·32-s − 10·34-s − 35-s − 4·37-s − 2·43-s + 10·44-s − 2·46-s + 9·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 0.632·10-s + 1.50·11-s + 0.832·13-s + 0.534·14-s − 16-s + 1.21·17-s + 0.447·20-s − 2.13·22-s + 0.208·23-s + 1/5·25-s − 1.17·26-s − 0.377·28-s − 0.557·29-s + 1.07·31-s + 1.41·32-s − 1.71·34-s − 0.169·35-s − 0.657·37-s − 0.304·43-s + 1.50·44-s − 0.294·46-s + 1.31·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.335019808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.335019808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.218696105737792327165744015083, −7.20882443661351012237272301509, −6.78064723068967301505822724764, −6.06211475047176405136233185943, −5.31244034452176443980496173680, −4.14207139243206784045574368093, −3.51994241316591159679412085246, −2.38574895562555243260750958943, −1.37725540122750241311441136605, −0.841090948221471581329869779220,
0.841090948221471581329869779220, 1.37725540122750241311441136605, 2.38574895562555243260750958943, 3.51994241316591159679412085246, 4.14207139243206784045574368093, 5.31244034452176443980496173680, 6.06211475047176405136233185943, 6.78064723068967301505822724764, 7.20882443661351012237272301509, 8.218696105737792327165744015083
