| L(s) = 1 | + 2.69·2-s + 3-s + 5.24·4-s + 5-s + 2.69·6-s − 0.797·7-s + 8.73·8-s + 9-s + 2.69·10-s + 2.59·11-s + 5.24·12-s + 2.79·13-s − 2.14·14-s + 15-s + 13.0·16-s − 5.77·17-s + 2.69·18-s + 5.24·20-s − 0.797·21-s + 6.97·22-s − 3.10·23-s + 8.73·24-s + 25-s + 7.52·26-s + 27-s − 4.18·28-s + 4.79·29-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 0.577·3-s + 2.62·4-s + 0.447·5-s + 1.09·6-s − 0.301·7-s + 3.08·8-s + 0.333·9-s + 0.851·10-s + 0.781·11-s + 1.51·12-s + 0.775·13-s − 0.573·14-s + 0.258·15-s + 3.25·16-s − 1.40·17-s + 0.634·18-s + 1.17·20-s − 0.173·21-s + 1.48·22-s − 0.647·23-s + 1.78·24-s + 0.200·25-s + 1.47·26-s + 0.192·27-s − 0.789·28-s + 0.889·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.919337310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.919337310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 7 | \( 1 + 0.797T + 7T^{2} \) |
| 11 | \( 1 - 2.59T + 11T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 17 | \( 1 + 5.77T + 17T^{2} \) |
| 23 | \( 1 + 3.10T + 23T^{2} \) |
| 29 | \( 1 - 4.79T + 29T^{2} \) |
| 31 | \( 1 - 9.48T + 31T^{2} \) |
| 37 | \( 1 + 7.69T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 + 2.79T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 8.87T + 53T^{2} \) |
| 59 | \( 1 + 1.08T + 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 8.50T + 73T^{2} \) |
| 79 | \( 1 + 6.48T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.138525380168549397938070725807, −6.83880392662720164780696900962, −6.64121121842086762463026515304, −6.08094901000050350774999152360, −5.05617677625977594549169227075, −4.46833573210813958631462871175, −3.71971767275104789155218137542, −3.09725519347025551541663215355, −2.22627011020930451794773391698, −1.46874180672732826845638630457,
1.46874180672732826845638630457, 2.22627011020930451794773391698, 3.09725519347025551541663215355, 3.71971767275104789155218137542, 4.46833573210813958631462871175, 5.05617677625977594549169227075, 6.08094901000050350774999152360, 6.64121121842086762463026515304, 6.83880392662720164780696900962, 8.138525380168549397938070725807
