L(s) = 1 | + 3.41·3-s + 0.861·5-s + 4.65·7-s + 8.68·9-s + 1.32·11-s − 1.15·13-s + 2.94·15-s − 4.38·17-s − 19-s + 15.9·21-s − 2.98·23-s − 4.25·25-s + 19.4·27-s + 2.42·29-s − 9.30·31-s + 4.54·33-s + 4.01·35-s + 12.1·37-s − 3.94·39-s + 8.84·41-s − 5.95·43-s + 7.48·45-s − 3.44·47-s + 14.6·49-s − 15.0·51-s − 10.8·53-s + 1.14·55-s + ⋯ |
L(s) = 1 | + 1.97·3-s + 0.385·5-s + 1.76·7-s + 2.89·9-s + 0.400·11-s − 0.319·13-s + 0.760·15-s − 1.06·17-s − 0.229·19-s + 3.47·21-s − 0.622·23-s − 0.851·25-s + 3.73·27-s + 0.450·29-s − 1.67·31-s + 0.791·33-s + 0.678·35-s + 1.99·37-s − 0.631·39-s + 1.38·41-s − 0.907·43-s + 1.11·45-s − 0.502·47-s + 2.09·49-s − 2.10·51-s − 1.48·53-s + 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.939530653\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.939530653\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 - 3.41T + 3T^{2} \) |
| 5 | \( 1 - 0.861T + 5T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 23 | \( 1 + 2.98T + 23T^{2} \) |
| 29 | \( 1 - 2.42T + 29T^{2} \) |
| 31 | \( 1 + 9.30T + 31T^{2} \) |
| 37 | \( 1 - 12.1T + 37T^{2} \) |
| 41 | \( 1 - 8.84T + 41T^{2} \) |
| 43 | \( 1 + 5.95T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 6.73T + 59T^{2} \) |
| 61 | \( 1 + 1.08T + 61T^{2} \) |
| 67 | \( 1 + 7.05T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 4.58T + 73T^{2} \) |
| 83 | \( 1 + 3.57T + 83T^{2} \) |
| 89 | \( 1 + 4.58T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.137361833233229577109633701440, −7.66342560539402689778148631764, −7.00056516468723427918694727217, −6.01442643426182308675581906398, −4.82879266540481792072807167005, −4.34345578701107553195939475103, −3.68504802665785978344503655533, −2.48871728560001214762100676515, −2.04471038064033263200764390121, −1.36227098193342456270264080733,
1.36227098193342456270264080733, 2.04471038064033263200764390121, 2.48871728560001214762100676515, 3.68504802665785978344503655533, 4.34345578701107553195939475103, 4.82879266540481792072807167005, 6.01442643426182308675581906398, 7.00056516468723427918694727217, 7.66342560539402689778148631764, 8.137361833233229577109633701440