| L(s) = 1 | + 3·3-s − 14·5-s + 9·9-s + 28·11-s + 74·13-s − 42·15-s − 82·17-s + 92·19-s − 8·23-s + 71·25-s + 27·27-s − 138·29-s + 80·31-s + 84·33-s + 30·37-s + 222·39-s − 282·41-s − 4·43-s − 126·45-s + 240·47-s − 246·51-s − 130·53-s − 392·55-s + 276·57-s + 596·59-s + 218·61-s − 1.03e3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.25·5-s + 1/3·9-s + 0.767·11-s + 1.57·13-s − 0.722·15-s − 1.16·17-s + 1.11·19-s − 0.0725·23-s + 0.567·25-s + 0.192·27-s − 0.883·29-s + 0.463·31-s + 0.443·33-s + 0.133·37-s + 0.911·39-s − 1.07·41-s − 0.0141·43-s − 0.417·45-s + 0.744·47-s − 0.675·51-s − 0.336·53-s − 0.961·55-s + 0.641·57-s + 1.31·59-s + 0.457·61-s − 1.97·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.328047504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328047504\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 74 T + p^{3} T^{2} \) |
| 17 | \( 1 + 82 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 8 T + p^{3} T^{2} \) |
| 29 | \( 1 + 138 T + p^{3} T^{2} \) |
| 31 | \( 1 - 80 T + p^{3} T^{2} \) |
| 37 | \( 1 - 30 T + p^{3} T^{2} \) |
| 41 | \( 1 + 282 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 130 T + p^{3} T^{2} \) |
| 59 | \( 1 - 596 T + p^{3} T^{2} \) |
| 61 | \( 1 - 218 T + p^{3} T^{2} \) |
| 67 | \( 1 - 436 T + p^{3} T^{2} \) |
| 71 | \( 1 + 856 T + p^{3} T^{2} \) |
| 73 | \( 1 - 998 T + p^{3} T^{2} \) |
| 79 | \( 1 - 32 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1508 T + p^{3} T^{2} \) |
| 89 | \( 1 - 246 T + p^{3} T^{2} \) |
| 97 | \( 1 + 866 T + p^{3} T^{2} \) |
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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
−8.579055416027817435565439684524, −8.007959174207763067406821117941, −7.14354651118487891619686248633, −6.53342514145545043578012937823, −5.46829628561178867013303542832, −4.24225935916204330688653138803, −3.83727129365002014617488461490, −3.06301178362921900413731501975, −1.72345783502553920137410545392, −0.68181462913399906938096525908,
0.68181462913399906938096525908, 1.72345783502553920137410545392, 3.06301178362921900413731501975, 3.83727129365002014617488461490, 4.24225935916204330688653138803, 5.46829628561178867013303542832, 6.53342514145545043578012937823, 7.14354651118487891619686248633, 8.007959174207763067406821117941, 8.579055416027817435565439684524
