| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4.82·7-s − 8-s + 9-s + 4.82·11-s + 12-s − 0.828·13-s + 4.82·14-s + 16-s − 4.82·17-s − 18-s − 4.82·21-s − 4.82·22-s − 8.48·23-s − 24-s + 0.828·26-s + 27-s − 4.82·28-s − 1.65·29-s + 31-s − 32-s + 4.82·33-s + 4.82·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.82·7-s − 0.353·8-s + 0.333·9-s + 1.45·11-s + 0.288·12-s − 0.229·13-s + 1.29·14-s + 0.250·16-s − 1.17·17-s − 0.235·18-s − 1.05·21-s − 1.02·22-s − 1.76·23-s − 0.204·24-s + 0.162·26-s + 0.192·27-s − 0.912·28-s − 0.307·29-s + 0.179·31-s − 0.176·32-s + 0.840·33-s + 0.828·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.138638233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.138638233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 37 | \( 1 - 6.48T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.581636280308737745359761919856, −7.57985465836185020984000911058, −6.79178422923294999197714525918, −6.46036711631492471800898869994, −5.72219862046473354669604351960, −4.11943069210627367095219913145, −3.80405667862955509188686189914, −2.74097713763147365275520572757, −2.00492020266272624465111251592, −0.61651545194429959382081252573,
0.61651545194429959382081252573, 2.00492020266272624465111251592, 2.74097713763147365275520572757, 3.80405667862955509188686189914, 4.11943069210627367095219913145, 5.72219862046473354669604351960, 6.46036711631492471800898869994, 6.79178422923294999197714525918, 7.57985465836185020984000911058, 8.581636280308737745359761919856
